Knowledge that Transforms

Abstract : Recent proposals by Gomory and others for solving linear programs involving integer-valued variables appear sufficiently promising that it is worthwhile to systematically review and classify problems that can be reduced to this class and thereby solved. Historically, non-linear, nonconvex and combinatorial problems are areas where classical mathematics almost always fails. It is therefore significant that the reduction can be made for problems involving multiple dichotomies and k-fold alternatives which include problems with discrete variables, non-linear separable minimizing functions, conditional constraints, global minimum of general concave functions and combinatorial problems such as the fixed charge problem, traveling salesman problem, orthogonal latin square problems, and map coloring problems.

THIS PAPER PRESENTS, on the assumption of lognormality, a simple graphical method of deriving the Engel elasticities for various items of consumer expenditure from what are known as concentration curves. Usually two types of such curves are distinguished: (a) the Lorenz curve which relates the proportion of total expenditure to the proportion of persons spending up to a given level of total expenditure per capita, and (b) the specific concentration curve which relates the proportion of total consumption of a specific commodity to the proportion of persons spending up to a given level of total expenditure per capita. We shall briefly indicate a method of using these curves to calculate the Engel elasticities and shall present some numerical results for a few important items of consumers' expenditure. These estimates are compared with other estimates obtained by the conventional method of least squares, which requires additional computations.

This paper investigates Bohm-Bawerk's idea of a preference for advancing the timing of future satisfactions from a somewhat different point of view. It is shown that simple postulates about the utility function of a consumption program for an infinite future logically imply impatience at least for certain broad classes of programs. The postulates assert continuity, sensitivity, stationarity of the utility function, the absence of intertemporal complementarity, and the existence of a best and a worst program. The more technical parts of the proof are set off in starred sections.